Commutative diagram

[2] A commutative diagram often consists of three parts: In algebra texts, the type of morphism can be denoted with different arrow usages: The meanings of different arrows are not entirely standardized: the arrows used for monomorphisms, epimorphisms, and isomorphisms are also used for injections, surjections, and bijections, as well as the cofibrations, fibrations, and weak equivalences in a model category.

In the left diagram, which expresses the first isomorphism theorem, commutativity of the triangle means that

In order for the diagram below to commute, three equalities must be satisfied: Here, since the first equality follows from the last two, it suffices to show that (2) and (3) are true in order for the diagram to commute.

Diagram chasing (also called diagrammatic search) is a method of mathematical proof used especially in homological algebra, where one establishes a property of some morphism by tracing the elements of a commutative diagram.

[5] A syllogism is constructed, for which the graphical display of the diagram is just a visual aid.

In this setting, commutative diagrams may include these higher arrows as well, which are often depicted in the following style:

As a simple example, the diagram of a single object with an endomorphism (

, sometimes called the free quiver), as used in the definition of equalizer need not commute.

Further, diagrams may be messy or impossible to draw, when the number of objects or morphisms is large (or even infinite).